The effect of the presence of substrates below metal grids on light transmission is investigated through finite-different time-domain (FDTD) simulations. Comparing grids on substrates with suspended grids, we identify the effects of the presence of substrates on the transmittances of metal grids. The presence of substrates below micron-scale grids has no specific effect on their transmittances; however, unexpected dips and flattened peaks in transmission spectra were observed in nano-scale grids. The figures of merits (FoMs) of metal grids are calculated using estimated transmittances and grid sheet resistances. Due to their lower resistances and higher transmittances, micron-scale grids show higher FoMs than nano-scale grids and, are thus promising transparent conducting electrode candidates. The best 1D grid electrode in this work exhibited a figure of merit, σdc/σop, > 1000
© 2014 Optical Society of America
Transparent conductive electrodes (TCE) are key components in optoelectronic devices, such as flat panel displays, light emitting diodes, photovoltaic devices, and touch screens [1–3]. Indium tin oxide (ITO) has commonly been used in industry as a transparent electrode because of its low electrical resistance and high optical transmittance in the visible region. However, the price of indium, the main element of ITO, has drastically increased owing to its depletion. ITO also has a few drawbacks, including a lack of flexibility  and a tendency to damage organic layers during sputtering , which could be critical problems for flexible displays and organic light emitting diodes (OLED). For these reasons, many studies have been carried out to develop next-generation transparent electrodes from conducting polymers [6, 7], graphene [8, 9], carbon nanotubes [10, 11], metal nanowires [12–15], and metal grids [1, 3, 16], as well as hybrid structures of these candidates [17, 18]. A recent review by Ellmer indicated that metal based TCEs are some of the most promising candidates for next-generation TCEs because of their high transmittances and low sheet resistances . Fan et al. demonstrated that high aspect ratio metal grids show the highest transmittances with affordable sheet resistances . Van de Groep et al. demonstrated how the widths of silver grid networks affect light transmission in metal nano-grids . However, these studies are limited to nano-scale metal grids with restricted period ranges and did not elucidate the influences of the periods on incident light transmission. In addition, the effects of the presence of substrates on the optical transmissions of metal grid transparent electrodes have not been reported.
In this work, we calculate the transmittances and sheet resistances of 1-D networks of metal grids on SiO2 substrates with linewidths ranging from 50 nm to 1 μm. Grid periods were fixed at 10 and 20 times longer than the linewidths (90% and 95% opening ratios) to attain high optical transparencies above 90%. By comparing the transmissions observed in the presence of substrates to those observed in the absence of substrates, the interactions between the substrates and the metal grids were investigated. Two types of metal (silver and aluminum) were employed to verify how the composition can affect the transmission of the metal grids.
2. Simulation method
A commercial FDTD (Finite-difference time-domain)  tool was employed to calculate the transmittance of 1D metal grids in a wavelength region of 300 to 900 nm. The optical constants for Ag and Al were taken from the experimental data in the Handbook of Chemistry and Physics . Perfectly matched layer (PML) boundary conditions were used for the upper and lower boundaries of the simulations. Periodic boundary conditions were applied to the x-axis boundaries to analyze the periodic structure of the grid. Transverse electric (TE: electric field parallel to the grids) and transverse magnetic (TM: magnetic field parallel to the grids) polarization can be analyzed independently. We used 90 and 95% opening ratio structures (the ratio of a period to the width is 10 and 20, respectively) maintaining metal thicknesses in 50 nm. Ag and Al were employed as grid materials with grid widths ranging from 50 to 1 μm. The metal grids were laid on 80 μm thick quartz substrates.
3. Results and discussion
3.1. Optical properties of grids
Rayleigh anomalies (RAs) and localized surface plasmon resonances (LSPRs) are generated in metal nano grids with transverse electric polarized light (TE-pol light) and transverse magnetic polarized light (TM-pol light), respectively [1, 3, 16]. Figure 1 presents the TE-pol light transmittance results calculated for nano-width grids. In the spectra of the suspended structures, there are sharp peaks caused by RAs (solid lines) [16, 21, 22]. Aluminum and silver grids with the same structure show RA peaks at the same position because peak position is determined by the period of the grid structure according to the following relationship:, where d is the grid period, θ is the incident angle, λ is the wavelength of incident light, and m is a positive integer . At normal incidence (sinθ is zero), the equation can be simplified to . For grids on substrates, RA peak positions are affected by the refractive indices of the substrates according to the following diffraction equation:, where n1 and n2 are the refractive indices of the atmospheres of the incident and diffracted lights, respectively, θ1 and θ2 are the angles of the incident and diffracted lights, respectively, λ is the wavelength of the light in vacuum, m is the order of diffraction, and d is the period of the grid. RA peaks occurs when sinθ2 = ± 1 and sinθ1 is zero, at normal incidence. Then, the diffraction condition can be simplified to. Comparing the two simple equations for normal incidence, the Rayleigh wavelengths obtained from the samples on the substrates were 1.5 times longer than those obtained from the samples without substrates.
In the structure, which has a 1 μm period (Fig. 1.,top left) peaks should, in theory, be generated at 333 and 500 nm for the suspended grid and 375, 500, and 750 nm (vertical dashed lines) for the same grid on a substrate (n = 1.5). Unlike the suspended structure, peaks obtained from the sample with a substrate were weak, and the spectrum not only showed insignificant peaks at 375 and 750 nm, but a slight dip at 500 nm.
Figure 2 exhibits an electric field profile around an Ag grid with a width of 50 nm and a period of 1 µm. When the Ag grid was suspended, a checkered electric field interference pattern was generated due to alternating constructive and destructive interference. These patterns can be obtained from suspended metal grids at RA wavelengths (TE propagating mode), and one previous study has suggested that constructive interference around grids increases their transmittances of light at RA wavelengths . This constructive interference does not occur when light is propagated through a substrate. It is speculated that this lack of constructive interference makes the RA peaks weak.
Figure 3 shows the calculated results for the TM-pol light transmittance of a nano-width grid. The transmission dip, which was not found in the transmission spectrum of the suspended structure, was generated by both the Ag and Al grids on substrates (dashed circles in Fig. 3). These dips could have been related to interactions between the substrates and localized surface plasmon resonances (LSPRs) because the dips did not shift when the grid periods were altered. The insets in Fig. 3 show the powers absorbed by the metal grids at the dips, as calculated using the divergences of the Poynting vectors. It is shown that the metal grids absorb strongly along their substrates. Comparing the integrated power absorptions in a single metal line, the 50 and 100 nm linewidth grids on the substrates exhibited absorptions 2.17 and 2.62 times higher than those in the suspended structures, respectively. It is known that strong damped (antisymmetric) short range surface plasmon polaritons occur and result in the extraordinary low transmission for thin Ag films with an asymmetric geometry (εair < εglass) [24, 25].
Figure 4 shows the average transmittance of each structure in a wavelength range of 300 to 900 nm, as a function of metal grid linewidth. “Substrate-referenced” denotes the transmittance that can be attributed to the grid alone, as determined by subtracting the transmittance of the reference quartz substrate from that of the grid on a substrate. For 500 nm and 1μm linewidth structures, transmittances obtained from the suspended grids were quite similar to those of the substrate-referenced for both 90 and 95% opening ratio structures. This indicates that the transmittance differences between the “on substrate” (black squares in Fig. 4) and the “without substrate” (red circles in Fig. 4) are caused by light loss from the substrate alone (6.45%), which suggests that there are no additional interactions between the substrates and the metal grids. Differences between the black squares and blue triangles in Fig. 4 were found in the 50 and 100 nm linewidth structures and were due to additional light loss resulting from interactions between the substrates and the LSPRs of the grids (dashed circles in Fig. 3).
The transmission of the 2-dimensional grid was also calculated. By combining the 1D grids of the 95% opening ratio structures with the same grids rotated by 90 degrees, a 2D grid was generated with an opening ratio of approximately 90%. Figure 5 shows the TE and TM-pol light transmission and light loss spectra of 2D and 1D grids. The light loss data demonstrates that the sum of the TE and TM light losses of the 1D grid were similar to the light loss of the 2D grid. Since the two 1D grids had the same opening ratio, the transmission of the 2D grid was expected to have the same transmission as can be calculated from the sum of the losses in the TE and TM in1D grids. However, an exceptional transmission dip, slightly deeper (few percent) than the sum of the TE and TM pol light losses of the 1D grid, was found at 500 nm in the spectrum of the 2D grid. The simulation of the 2D grid needs a 3D simulation region, which requires significant memory and time. However, using 1D grid simulations could reduce the time needed to estimate the transmissions of the 2D grids.
3.2 Electrical properties of the metal grids
Sheet resistances were calculated from the electrical resistivity of the bulk Ag and Al to be 3.2 and 6.3 Ω/□ for Ag grids and 5.6 and 11.3 Ω/□ for Al grids with 95% and 90% opening ratios, respectively. However, these values were found to be inaccurate. Previous research found that the actual resistivity of a similar Ag nanowire network was approximately 6 times greater than that of the bulk material because of structural defects, wire discontinuities, wire-width variations, and grain boundary scattering [26, 27]. For the same reasons, the resistivity of the aluminum nanowire (width: 24 nm, thickness: 15 nm, length: 224 μm) was approximately 7 times greater than the resistivity of the bulk Al . Considering these increases in resistivity (~7 times higher than the bulk resistivities), the sheet resistances of the 90 and 95% opening ratio samples were expected to be 22 Ω/□ and 44 Ω/□ for Ag and 40 Ω/□ and 80 Ω/□ for Al, respectively.
In view of the line resistivity model, increases in the resistivities caused by limiting widths in the micro-width grids can be neglected [29, 30] since these widths were more than 10 times larger than their electron mean free paths (Ag: 52 nm, Al: 15 nm) . The resistivities of 50 nm thick Ag and Al have been reported to be 1.6 times higher than those of the bulk materials . Hence, the sheet resistances of the micro-width grids with opening ratios of 90 and 95% were expected to be 5.1 Ω/□ and 10.2 Ω/□, respectively, for Ag and 9 Ω/□, 18Ω/□, respectively, for Al.
Using the expected sheet resistance values above, the figures of merits (FoMs) were calculated in order to compare their efficiencies as transparent electrode candidates. FoM was defined as the ratio of the electrical conductance to the optical conductance (σdc/σopt), which is given by, where the expected sheet resistance and averaged transmission were used for Rs and T, respectively . Figure 6 shows the FoMs of the Al and Ag grid electrodes plotted as functions of the linewidths of the metal grids. The σdc/σopt of ITO is approximately 120–240  and that of an Ag nanowire network, the leading candidate to replace ITO, is reported to be approximately 215 [35, 36]. Other materials including Al-Cu wire, graphene, and carbon nanotubes were reported to be approximately 75 [2, 34–36]. However, our result shows that the σdc/σopt values for micron grids are 600–1100 for Ag and 400–600 for Al. The FoM values for the Ag and Al nano-grids are 130–180 and 70-100, respectively. It should be noted that the high σdc/σopt values obtained from the micro width grids are mainly due to the decreased resistivities since the observed transmittance increases were trivial.
In conclusion, the 1D Ag and Al grids exhibit low sheet resistances and very high transmittances in the visible region. On substrates, the metal grids with 50 and 100 nm linewidths exhibit decreased transmittances due to weak RA peaks and transmission dips caused by interactions between LSPRs and substrates. Metal grids with 500 nm and 1 μm linewidths demonstrated uniform transmittance in the visible light range, corresponding to their opening ratios and low sheet resistances, similar to their bulk resistivities. 2D simulation results can be calculated from the light loss of the 1D grid in TE and TM-pol light. The substrates were considered in simulations because they contribute to the transmittances of the metal grids. To achieve lower resistances and higher transmittances, metal grids with larger linewidths (> 100 nm) should be used.
This research was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2009-0083540 and 2011-0028570)
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